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NNTP-Posting-Date: Sat, 21 Dec 2024 23:29:26 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 21 Dec 2024 15:29:14 -0800
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On 12/21/2024 11:32 AM, Jim Burns wrote:
> On 12/21/2024 6:34 AM, WM wrote:
>> On 20.12.2024 19:48, Jim Burns wrote:
>>> On 12/19/2024 4:37 PM, WM wrote:
>
>>>> That means all numbers are lost by loss of
>>>> one number per term.
>>>>
>>>> That implies finite endsegments.
>>>
>>> Q. What does 'finite' mean?
>
> Consider end.segments of the finite cardinals.
>
> Q. What does 'finite' mean,
> 'finite', whether darkᵂᴹ or visibleᵂᴹ?
>
>> Here is a new and better definition of endsegments
>>
>> E(n) = {n+1, n+2, n+3, ...} with E(0) = ℕ.
>>
>> ∀n ∈ ℕ : E(n+1) = E(n) \ {n+1}
>> means that the sequence of endsegments can decrease only by one
>> natnumber per step.
>
> E(n+1) is larger.than each of
> the sets for which there are smaller.by.one sets.
> E(n+1) isn't any of
> the sets for which there are smaller.by.one sets.
>
> E(n+1) isn't smaller.by.one than E(n).
> E(n+1) is emptier.by.one than E(n)
>
>> Therefore the sequence of endsegments
>> cannot become empty
>
> Yes, because
> the sequence of end.segments
> can become emptier.one.by.one, but
> it cannot become smaller.one.by.one.
>
>> (i.e., not all natnumbers can be applied as indices)
>
> Each finite.cardinal can be applied,
> which makes the sequence emptier.by.one
> but does not make the sequence smaller.by.one.
>
>> unless the empty endsegment is reached,
>
> Each end.segment is emptier.by.one.
> No end.segment is smaller.than the first end.segment ℕ
> The empty end.segment is not reached.
>
> No finite.cardinal is in common with
> all infinitely.many
> infinite.end.segments of
> finite.cardinals.
>
> Nothing other.than a finite.cardinal is in
> any end.segment of the finite.cardinals,
> or in their intersection.
>
> The intersection of all infinitely.many
> infinite end.segments of finite.cardinals
> is not an end.segment
> but is empty.
>
> Q. What does 'finite' mean?
>
>> unless the empty endsegment is reached,
>
> The empty end.segment, not.existing, is not.reached.
>
> The intersection.of.finitely.many is not.empty.
> The intersection.of.all is empty.
>
>> and
>> before finite endsegments,
>> endsegments containing only 1, 2, 3, or n ∈ ℕ numbers,
>> have been passed.
>
> ⎛ Assume end.segment E(n) of the finite.cardinals
> ⎜ holds only finite.cardinal.k.many finite.cardinals.
> ⎜
> ⎜ k.sized E(n) holds
> ⎜ kᵗʰ.smallest, 1ˢᵗ.largest finite.cardinal E(n)[k]
> ⎜
> ⎜ If a finite.cardinal larger.than E(n)[k] exists,
> ⎜ it would also be in end.segment E(n) and
> ⎜ larger.than 1ˢᵗ.largest E(n)[k]: a contradiction.
> ⎜
> ⎜ Thus,
> ⎜ a finite.cardinal larger.than E(n)[k] doesn't exist.
> ⎜
> ⎜ However,
> ⎜⎛ for each finite.cardinal j,
> ⎜⎝ larger.than.j finite.cardinal j+1 exists.
> ⎜
> ⎜ Larger.than.E(n)[k] finite.cardinal E(n)[k]+1 exists.
> ⎝ Contradiction.
>
> Therefore,
> end.segment E(n) of the finite.cardinals does not hold
> only finite.cardinal.many finite.cardinals.
>
> There are no finite end.segments of the finite.cardinals.
>
> Q. What does 'finite' mean?
>
>> These however, if existing at all, cannot be seen.
>> They are dark.
>
> Darknessᵂᴹ and visibilityᵂᴹ don't change any of this.
> There are no finite end.segments of the finite.cardinals.
>
> We know it by the method of
> assembling finite sequences of claims (proofs),
> each claim of which is true.or.not.first.false (valid),
> and holding those claims.we.know (theorems),
> because
> a finite sequence of claims,
> each claim of which true.or.not.first.false,
> holds only true claims.
>
> Some claims (definitions)
> we know are true because
> we know how we have defined things.
>
> Some claims (valid inferences)
> we know are not.first.false because
> we can inspect the finite sequence of claims.
>
> None of _the claims_ are darkᵂᴹ,
> whatever the status of _what the claims are about_
>
> Darknessᵂᴹ or visibilityᵂᴹ of finite.cardinals
> don't change _the claims_
>
>>>> That means all numbers are lost by loss of
>>>> one number per term.
>>>>
>>>> That implies finite endsegments.
>>>
>>> No.
>>> Yes, each number is lost by loss of
>>> one number per term.
>>> However,
>>> each end.segment is not finite.
>>
>
>> Then the last endsegment is empty.
>
> There is no last end.segment of the finite.cardinals.
> ⎛ For each finite.cardinal j,
> ⎝ larger.than.j finite.cardinal j+1 exists.
> contradicts a last end.segment.
>
>

Well, anybody can just build "infinite-middle",
is what it is.

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